In order to construct good moduli spaces for vector bundles over algebraic curves, Mumford introduced the concept of a stable vector bundle. This concept has been generalized to vector bundles and, more generally, coherent sheaves over algebraic manifolds by Takemoto, Bogomolov and Gieseker. As the differential geometric counterpart to the stability, I introduced the concept of an Einstein-Hermitian vector bundle. The main purpose of this book is to lay a foundation for the theory of Einstein-Hermitian vector bundles. We shall not give a detailed introduction here in this preface since the table of contents is fairly self-explanatory and, furthermore, each chapter is headed by a brief introduction.
My first serious encounter with stable vector bundles was in the summer of 1978 in Bonn, when F. Sakai and M. Reid explained to me the work of Bogomolov on stable vector bundles. This has led me to the concept of an Einstein-Hermitian vector bundle. In the summer of 1981 when I met M. Lubk'e at DMV Seminar in Dusseldorf, he was completing the proof of his inequality for Einstein-Hermitian vector bundles, which rekindled my interest in the subject.
With this renewed interest, I lectured on vanishing theorems and Einstein-Hermitian vector bundles at the University of Tokyo in the fall of 1981. The notes taken by I. Enoki and published as Seminar Notes 41 of the Department of Mathematics of the University of Tokyo contained good part of Chapters I, III, IV and V of this book. Without his notes which filled in many details of my lectures, this writing project would not have started. In those lectures I placed
much emphasis------perhaps too much emphasis------on vanishing theorems. In
retrospect, we need mostly vanishing theorems for holomorphic sections for the purpose of this book, but I decided to include cohomology vanishing theorems as well.
During the academic year 1982/83 in Berkeley and in the summer of 1984 in Tsukuba, I gave a course on holomorphic vector bundles. The notes of these lectures ("Stable Vector Bundles and Curvature" in the "Survey in^Geometry" series) distributed to the audience, consisted of the better part of Chapters 1 through V. My lectures at the Tsukuba workshop were supplemented by talks by T. Mabuchi (on Donaldson's work) and by M. Itoh (on Yang-Mills theory). In writing Chapter VI, which is mainly on the work of Donaldson on stable bundles
